Integrand size = 11, antiderivative size = 121 \[ \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx=\frac {1}{2} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )+\frac {1}{2} \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right ) \]
1/2*arctan((3+cot(x)*(1-2^(1/2))-2*2^(1/2))/(1+cot(x))^(1/2)/(-14+10*2^(1/ 2))^(1/2))*(2^(1/2)-1)^(1/2)+1/2*arctanh((3+2*2^(1/2)+cot(x)*(1+2^(1/2)))/ (1+cot(x))^(1/2)/(14+10*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.42 \[ \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )}{\sqrt {1-i}}+\frac {\text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )}{\sqrt {1+i}} \]
ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]]/Sqrt[1 - I] + ArcTanh[Sqrt[1 + Cot[x ]]/Sqrt[1 + I]]/Sqrt[1 + I]
Time = 0.40 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 25, 4019, 25, 3042, 4018, 216, 220}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot (x)}{\sqrt {\cot (x)+1}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4019 |
| \(\displaystyle \frac {\int -\frac {1-\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {\int -\frac {1-\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {1-\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {1-\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1-\left (-1+\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}-\frac {\int \frac {1-\left (-1-\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}\) |
| \(\Big \downarrow \) 4018 |
| \(\displaystyle \frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3\right )^2}{\cot (x)+1}-2 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {\cot (x)+1}}\right )}{\sqrt {2}}-\frac {\left (3-2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3\right )^2}{\cot (x)+1}-2 \left (7-5 \sqrt {2}\right )}d\left (-\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {\cot (x)+1}}\right )}{\sqrt {2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3\right )^2}{\cot (x)+1}-2 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {\cot (x)+1}}\right )}{\sqrt {2}}+\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7}}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7}}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\cot (x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}}}\) |
((3 - 2*Sqrt[2])*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])])/(2*Sqrt[-7 + 5*Sqrt[2]]) + ((3 + 2*Sqrt [2])*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqrt[2] )]*Sqrt[1 + Cot[x]])])/(2*Sqrt[7 + 5*Sqrt[2]])
3.1.46.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[-2*(d^2/f) Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0 ] && NeQ[c^2 + d^2, 0] && EqQ[2*a*c*d - b*(c^2 - d^2), 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( f_.)*(x_)]], x_Symbol] :> With[{q = Rt[a^2 + b^2, 2]}, Simp[1/(2*q) Int[( a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x], x] - Simp[1/(2*q) Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f *x])/Sqrt[a + b*Tan[e + f*x]], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && N eQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2*a*c*d - b*(c^2 - d^2), 0] && NiceSqrtQ[a^2 + b^2]
Leaf count of result is larger than twice the leaf count of optimal. \(180\) vs. \(2(85)=170\).
Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.50
| method | result | size |
| derivativedivides | \(\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}+\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}\) | \(181\) |
| default | \(\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}+\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}\) | \(181\) |
1/4*2^(1/2)*(-1/2*(2+2*2^(1/2))^(1/2)*ln(1+cot(x)+2^(1/2)-(1+cot(x))^(1/2) *(2+2*2^(1/2))^(1/2))+2*(1-2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+cot( x))^(1/2)-(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2)))+1/4*2^(1/2)*(1/2*(2+ 2*2^(1/2))^(1/2)*ln(1+cot(x)+2^(1/2)+(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2)) +2*(1-2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+cot(x))^(1/2)+(2+2*2^(1/2 ))^(1/2))/(-2+2*2^(1/2))^(1/2)))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.30 \[ \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx=\frac {1}{4} \, \sqrt {2} \sqrt {i + 1} \log \left (-\left (i - 1\right ) \, \sqrt {2} \sqrt {i + 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {i + 1} \log \left (\left (i - 1\right ) \, \sqrt {2} \sqrt {i + 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-i + 1} \log \left (\left (i + 1\right ) \, \sqrt {2} \sqrt {-i + 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-i + 1} \log \left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {-i + 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) \]
1/4*sqrt(2)*sqrt(I + 1)*log(-(I - 1)*sqrt(2)*sqrt(I + 1) + 2*sqrt((cos(2*x ) + sin(2*x) + 1)/sin(2*x))) - 1/4*sqrt(2)*sqrt(I + 1)*log((I - 1)*sqrt(2) *sqrt(I + 1) + 2*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) + 1/4*sqrt(2)*s qrt(-I + 1)*log((I + 1)*sqrt(2)*sqrt(-I + 1) + 2*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) - 1/4*sqrt(2)*sqrt(-I + 1)*log(-(I + 1)*sqrt(2)*sqrt(-I + 1) + 2*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x)))
\[ \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx=\int \frac {\cot {\left (x \right )}}{\sqrt {\cot {\left (x \right )} + 1}}\, dx \]
\[ \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx=\int { \frac {\cot \left (x\right )}{\sqrt {\cot \left (x\right ) + 1}} \,d x } \]
\[ \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx=\int { \frac {\cot \left (x\right )}{\sqrt {\cot \left (x\right ) + 1}} \,d x } \]
Time = 12.87 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.90 \[ \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx=\mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{128\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}-8}-\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{128\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}-8}\right )\,\left (2\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}+2\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}\right )-\mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{128\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}+8}+\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{128\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}+8}\right )\,\left (2\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}-2\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}\right ) \]
atanh((16*2^(1/2)*(1/16 - 2^(1/2)/16)^(1/2)*(cot(x) + 1)^(1/2))/(128*(1/16 - 2^(1/2)/16)^(1/2)*(2^(1/2)/16 + 1/16)^(1/2) - 8) - (16*2^(1/2)*(2^(1/2) /16 + 1/16)^(1/2)*(cot(x) + 1)^(1/2))/(128*(1/16 - 2^(1/2)/16)^(1/2)*(2^(1 /2)/16 + 1/16)^(1/2) - 8))*(2*(1/16 - 2^(1/2)/16)^(1/2) + 2*(2^(1/2)/16 + 1/16)^(1/2)) - atanh((16*2^(1/2)*(1/16 - 2^(1/2)/16)^(1/2)*(cot(x) + 1)^(1 /2))/(128*(1/16 - 2^(1/2)/16)^(1/2)*(2^(1/2)/16 + 1/16)^(1/2) + 8) + (16*2 ^(1/2)*(2^(1/2)/16 + 1/16)^(1/2)*(cot(x) + 1)^(1/2))/(128*(1/16 - 2^(1/2)/ 16)^(1/2)*(2^(1/2)/16 + 1/16)^(1/2) + 8))*(2*(1/16 - 2^(1/2)/16)^(1/2) - 2 *(2^(1/2)/16 + 1/16)^(1/2))